We present a novel Monte Carlo-based fluid simulation approach capable of pointwise and stochastic estimation of fluid motion. Drawing on the Feynman-Kac representation of the vorticity transport equation, we propose a recursive Monte Carlo estimator of the Biot-Savart law and extend it with a stream function formulation that allows us to treat free-slip boundary conditions using a Walk-on-Spheres algorithm. Inspired by the Monte Carlo literature in rendering, we design and compare variance reduction schemes suited to a fluid simulation context for the first time, show its applicability to complex boundary settings, and detail a simple and practical implementation with temporal grid caching. We validate the correctness of our approach via quantitative and qualitative evaluations - across a range of settings and domain geometries - and thoroughly explore its parameters' design space. Finally, we provide an in-depth discussion of several axes of future work building on this new numerical simulation modality.

Designing robust mutation strategies for primary sample space Metropolis light transport is a challenging problem: poorly tuned mutations both hinder state space exploration and introduce structured image artifacts. Scenes with complex materials, lighting, and geometry make hand-designing strategies that remain optimal over the entire state space infeasible. Moreover, these difficult regions are often sparse in state space, and so relying exclusively on intricate—and often expensive—proposal mechanisms can be wasteful, whereas simpler inexpensive mechanisms are more sample efficient. We generalize Metropolis–Hastings light transport to employ a flexible two-stage mutation strategy based on delayed rejection Markov chain Monte Carlo. Our approach generates multiple proposals based on the failure of previous ones, all while preserving Markov chain ergodicity. This allows us to reduce error while maintaining fast global exploration and low correlation across chains. Direct application of delayed rejection to light transport leads to low acceptance probabilities, and so we also propose a novel transition kernel to alleviate this issue. We benchmark our approach on several applications including bold-then-timid and cheap-then-expensive proposals across different light transport algorithms. Our method is applicable to any primary sample space algorithm with minimal implementation effort, producing consistently better results on a variety of challenging scenes.

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